Statistical Mechanics of Emergent Geometry in Random Graphs

Abstract

How spacetime emerges from featureless nothingness is one of the most intriguing questions in fundamental physics. In this thesis, we take on the random geometry approach to study discretized spacetime and follow the assumption that, in the sim- plest form, geometric structures may arise from random connections between dots under certain rules. We study a family of random graph models called Exponential Random Graph Models (ERGMs). Although this family was extensively investigated in the network science community as a proxy to study real-world social networks, its strength is in its formulation as a Gibbs-Boltzmann distribution in equilibrium sta- tistical mechanics. Thus, one can modify, analyze, and simulate the ensemble using familiar tools from statistical mechanics. We are interested in modifying the basic ERGMs to arrive at a random graph model that possesses emergent geometric properties. This would serve as a proof-of-principle that geometric spacetime may emerge from randomly connected dots. Our study leads to novel classes of random graphs whose edges can self-assemble themselves into both simple geometric primitives (e.g. triangles) and more complex structures (e.g. hexagons). The number of such structures is relatively large compared to the amount of dots and connections available in the graph. Lastly, but interestingly, our model is free from the graph collapse problem that is often observed in the traditional ERGMs.